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Fearless Symmetry: Exposing the Hidden Patterns of Numbers
Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them. Hidden symmetries were first discovered nearly two hundred years ago by French mathematician évariste Galois. They have been used extensively in the oldest and largest branch of mathematics--number theory--for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination. The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.
- GenresMathematicsScience
312 pages, Kindle Edition
First published May 22, 2006
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June 19, 2020
La intención de los autores de este texto es presentar «con un estilo amigable para una audiencia general» (primera fase de la cuarta de forros) una discusión sobre patrones numéricos y técnicas empleadas por los matemáticos para hallarlos. Desafortunadamente la afirmación de la que se parte y la buena intención de los autores no alcanzan para llevar al «gran público» un propósito tan vago.
El libro parte de las propiedades básicas de los números enteros y de los grupos de permutaciones. En algún momento se menciona el último teorema de Fermat-Wiles, sin embargo, poco a poco el texto se decanta más hacia la teorÃa de representaciones, las leyes de reciprocidad, y las curvas elÃpticas (tres ingredientes que son indispensables par la prueba de Wiles) y uno puede sentir cómo —por más que el esfuerzo de los autores haya sido denodadoâ€� la «audiencia general» se va quedando en el camino.
No digo que sea un libro “maloâ€�, al contrario, me parece excelente si (de menos) tienes una licenciatura en matemáticas o fÃsica, porque de lo contrario es muy fácil perderse en las discusiones donde los autores, hablando de la acción del grupo de Galois sobre la n-torsión de una curva elÃptica (por ejemplo), deciden, en aras de un estilo más amigable, no usar fórmulas.
¡Claro! Si estás interesado en las matemáticas, estás cursando la carrera, o solo te interesa acercarte a ella para ver qué cosas nuevas podrÃas aprender, entonces este libro podrÃa interesarte, aunque sea poco probable que obtengas el mayor beneficio de él o bien tengas que dejarlo incompleto en el camino.
Fuera de lo anterior, el libro pone sobre la mesa cómo las leyes de reciprocidad en teorÃa de números y la acción del grupo de Galois (del campo de TODOS los números algebraicos) sobre ciertas curvas de tercer grado guardan una estrecha relación que le permite a los profesionales de las matemáticas probar teoremas, y decir cosas interesantes sobre algunos sistemas de ecuaciones con coeficientes enteros.
Por último, si bien para esta nueva edición se eliminaron por completo casi todas las erratas, hay un par que sobrevivieron, pero son tan obvias que no le causarán confusión a nadie y es fácil detectarlas en el cuerpo del texto.
El libro parte de las propiedades básicas de los números enteros y de los grupos de permutaciones. En algún momento se menciona el último teorema de Fermat-Wiles, sin embargo, poco a poco el texto se decanta más hacia la teorÃa de representaciones, las leyes de reciprocidad, y las curvas elÃpticas (tres ingredientes que son indispensables par la prueba de Wiles) y uno puede sentir cómo —por más que el esfuerzo de los autores haya sido denodadoâ€� la «audiencia general» se va quedando en el camino.
No digo que sea un libro “maloâ€�, al contrario, me parece excelente si (de menos) tienes una licenciatura en matemáticas o fÃsica, porque de lo contrario es muy fácil perderse en las discusiones donde los autores, hablando de la acción del grupo de Galois sobre la n-torsión de una curva elÃptica (por ejemplo), deciden, en aras de un estilo más amigable, no usar fórmulas.
¡Claro! Si estás interesado en las matemáticas, estás cursando la carrera, o solo te interesa acercarte a ella para ver qué cosas nuevas podrÃas aprender, entonces este libro podrÃa interesarte, aunque sea poco probable que obtengas el mayor beneficio de él o bien tengas que dejarlo incompleto en el camino.
Fuera de lo anterior, el libro pone sobre la mesa cómo las leyes de reciprocidad en teorÃa de números y la acción del grupo de Galois (del campo de TODOS los números algebraicos) sobre ciertas curvas de tercer grado guardan una estrecha relación que le permite a los profesionales de las matemáticas probar teoremas, y decir cosas interesantes sobre algunos sistemas de ecuaciones con coeficientes enteros.
Por último, si bien para esta nueva edición se eliminaron por completo casi todas las erratas, hay un par que sobrevivieron, pero son tan obvias que no le causarán confusión a nadie y es fácil detectarlas en el cuerpo del texto.
September 8, 2015
This is a greatly illuminating book for those of us who find ourselves jumping around and skimming through Wikipedia articles on topics in algebra and number theory: fields, groups, Galois theory, quadratic reciprocity, elliptic curves, p-torsion points, field extensions, Frobenius, Etale cohomology etc. Reading this book will not impart a complete understanding of these topics--since the book is intended for laypersons--but it does give one an intuitive feel for the kinds of objects used in modern mathematics.
The first part is on Algebra (fields, groups, permutation groups, varieties, quadratic reciprocity) and is quite easy to follow with only a high-school math background.
The second is on Galois theory and representations (Permutations of roots of polynomials, absolute Galois group of Q, group law on elliptic curves, torsion points, Mod p Linear representations of the Absolute Galois, Galois group of a polynomial, restriction morphism, traces, Conjugacy classes, character of a linear representation, Frobenius, ramified primes, Algebraic integers, discriminant, norm). I wasn't familiar with most of these topics and spent the bulk of my time reading the book trying to understand them. I had to stop reading the book like a novel and had to reread many parts of it, sometimes multiple times. The good part is that the authors give simple illustrative examples for each newly introduced concept. Definitions and statements given in this part are graspable, and I think a layperson can get a reasonably complete picture of the objects introduced.
The third part is called Reciprocity Laws (traces of Frobenius, Frob_q on roots of Unity, one-dimensional Galois representations, 2-D reps from p-Torsion points of Elliptic curves, Linear action of groups, Linearization, Etale Cohomology, Modular forms, Fermat's Last theorem). Although the book builds up to sketching a proof of FLT, the first and the second part make enjoyable self-contained readings on their own. The third part begins well enough with the 1D and 2D Galois representations with easy-to-follow examples, but gets quite hazy when Etale Cohomology makes its entrance. Modular forms, Frey curves, and whatever else goes into proving FLT was just a blur and I don't think it told me any cohesive story that I will remember. I don't blame the authors though because omitting the hazy part would make the book anti-climactic, and fully covering it would make the book a formidable 1000-page monster that no one would get through.
This book is a great introductory book on algebra for people who aren't afraid to parse equations. It is more technical and precise than Simon Singh's FLT or other popular books, and hence also more concise. If you find popular pop-sci and pop-math books oversimplified or verbose, you'll find Fearless Symmetry delightful.
The first part is on Algebra (fields, groups, permutation groups, varieties, quadratic reciprocity) and is quite easy to follow with only a high-school math background.
The second is on Galois theory and representations (Permutations of roots of polynomials, absolute Galois group of Q, group law on elliptic curves, torsion points, Mod p Linear representations of the Absolute Galois, Galois group of a polynomial, restriction morphism, traces, Conjugacy classes, character of a linear representation, Frobenius, ramified primes, Algebraic integers, discriminant, norm). I wasn't familiar with most of these topics and spent the bulk of my time reading the book trying to understand them. I had to stop reading the book like a novel and had to reread many parts of it, sometimes multiple times. The good part is that the authors give simple illustrative examples for each newly introduced concept. Definitions and statements given in this part are graspable, and I think a layperson can get a reasonably complete picture of the objects introduced.
The third part is called Reciprocity Laws (traces of Frobenius, Frob_q on roots of Unity, one-dimensional Galois representations, 2-D reps from p-Torsion points of Elliptic curves, Linear action of groups, Linearization, Etale Cohomology, Modular forms, Fermat's Last theorem). Although the book builds up to sketching a proof of FLT, the first and the second part make enjoyable self-contained readings on their own. The third part begins well enough with the 1D and 2D Galois representations with easy-to-follow examples, but gets quite hazy when Etale Cohomology makes its entrance. Modular forms, Frey curves, and whatever else goes into proving FLT was just a blur and I don't think it told me any cohesive story that I will remember. I don't blame the authors though because omitting the hazy part would make the book anti-climactic, and fully covering it would make the book a formidable 1000-page monster that no one would get through.
This book is a great introductory book on algebra for people who aren't afraid to parse equations. It is more technical and precise than Simon Singh's FLT or other popular books, and hence also more concise. If you find popular pop-sci and pop-math books oversimplified or verbose, you'll find Fearless Symmetry delightful.
December 27, 2020
I loved reading it. I felt it gave me an inkling of what the hell class field theory and the Langlands program is trying to do. Unfortunately, the pacing was pretty off. It starts by assuming you know nothing, and ends at Artinian reciprocity; I was lost 3/4th of the way through. It's hard to articulate why I got lost --- it felt like piling up intuition on intuition led to a sandpile that collapsed at some point. I'll have to try reading the book again now and see if it reads better.
Regardless, I loved the experience.
Regardless, I loved the experience.
November 13, 2017
Kind of fun, but not quite enough to motivate me to code up the examples. It's definitely a departure from the math engineers learn. I did enjoy getting a glimpse into areas of mathematics I had no idea about. Definitely not a breezy read, especially the latter chapters where the authors seem to expect you to understand everything. It gets easier toward the very end where Wiles's proof of Fermat's Last Theorem is explained in very general terms. I didn't get as much of a sense of the beautiful symmetry of numbers out of it as the authors seemed to think I would. That may require a more careful reading that I was willing to give it.
May 13, 2015
This is supposed to popular science book, but after the 4th-5th chapter someone who is not familiar with advanced level math can't understand most of the topics, and since it's not textbook and many terms and topics are left unexplained, it's hard to read even if you are mathematician.
But, this book is nice try to summarize and emphasise importance of Galois theory, so I enjoyed reading (first half) of the book.
But, this book is nice try to summarize and emphasise importance of Galois theory, so I enjoyed reading (first half) of the book.
July 28, 2011
This book would be great for novices and those wanting a broad understanding into Hidden Patterns of Numbers. It does an incredible job in making the subjects it describes easy to understand to those who do not have a technical training in mathematics or physics.
April 11, 2023
Fearless Symmetry is an excellent book that provides a comprehensible introduction to the complex world of number theory, making it an ideal choice for readers who are not necessarily experts in mathematics. The author presents a thorough exploration of the mathematical concepts and techniques used to prove Fermat's Last Theorem, but does so in a way that is accessible and engaging to a broader audience. Fearless Symmetry is a well-written and insightful book that will be appreciated by anyone interested in the subject, as it showcases the beauty and elegance of mathematical reasoning.
January 18, 2020
I did not understand nearly as much in this book as I would have liked.
Looking over the table of contents I now realize that it was quite ambitious in trying to cover so much math for a general audience.
Looking over the table of contents I now realize that it was quite ambitious in trying to cover so much math for a general audience.
March 7, 2017
This is a strange book, sitting somewhere between a proper math text and a popular science book, without being either. The culminating bit, where Wiles' proof of FLT is sketched, is both dense in detail and sketchily handwavy. I'm not sure I came away with much from that part. And the first half of the book was too trivial for me. But some of the bits in between were interesting so it was still worth a read.
January 5, 2016
Inaccurate to describe this as a book in popular mathematics
A simple examination of the title and the cover leads you to believe that this is a book about the visual aspects of symmetry and how it is generated mathematically. However, there are only two images, one of a tetrahedron and the other of a sphere. The best way to describe the book is that it is a short and detailed journey through the mathematical background needed to explain representation theory, reciprocity rules, Galois theory and the basics of the proof of Fermat’s Last Theorem.
While a great deal of the book can be described by the phrase “popular book,� this is not a book on popular mathematics. The third and last part deals with topics that only people with a significant background in mathematics will understand. Other sections in parts one and two would also be difficult for the person not well schooled in mathematics to understand. In the foreword the authors recommend that the reader has studied calculus. I consider this too weak a background, the absolute minimum would be someone well-schooled in calculus.
If you are interested in learning a great deal of the background needed to understand the proof of Fermat’s Last Theorem and have some advanced mathematical background or are able to learn advanced mathematics on your own, this book will allow you to learn the necessary background. However, it will not be easy.
This book was made available for free for review purposes and this review also appears on Amazon
A simple examination of the title and the cover leads you to believe that this is a book about the visual aspects of symmetry and how it is generated mathematically. However, there are only two images, one of a tetrahedron and the other of a sphere. The best way to describe the book is that it is a short and detailed journey through the mathematical background needed to explain representation theory, reciprocity rules, Galois theory and the basics of the proof of Fermat’s Last Theorem.
While a great deal of the book can be described by the phrase “popular book,� this is not a book on popular mathematics. The third and last part deals with topics that only people with a significant background in mathematics will understand. Other sections in parts one and two would also be difficult for the person not well schooled in mathematics to understand. In the foreword the authors recommend that the reader has studied calculus. I consider this too weak a background, the absolute minimum would be someone well-schooled in calculus.
If you are interested in learning a great deal of the background needed to understand the proof of Fermat’s Last Theorem and have some advanced mathematical background or are able to learn advanced mathematics on your own, this book will allow you to learn the necessary background. However, it will not be easy.
This book was made available for free for review purposes and this review also appears on Amazon
June 10, 2016
Ash and Gross have written a hand-holding book for non-mathematicians, with absolute rigor, walking them through the Galois group and Galois theory (SYMMETRY of solutions over Fp of polynomial equations), as well as the mysterious and not-so-mysterious underpinnings of the Legendre symbol and quadratic reciprocity, in chapters 6 and 7 respectively.
The concept of varieties is introduced earlier.
After elliptic curves (Please note: I discovered Ash and Gross for their famous popular book on elliptic curves) are introduced in Chapter 9 (This is stand-alone, you dont need their other book), one is led through group representations and characters albeit briefly in Chapter 15, the complicated concepts of Frobenius elements are introduced in chapter 16.
Z-varieties and the reciprocity law in context of Frobenius are re-explained in the penultimate chapter.
The last chapter takes off in a way to show that math is a evolving area of research.
THe book boasts excellent credentials including a preface by Barry Mazur.
The concept of varieties is introduced earlier.
After elliptic curves (Please note: I discovered Ash and Gross for their famous popular book on elliptic curves) are introduced in Chapter 9 (This is stand-alone, you dont need their other book), one is led through group representations and characters albeit briefly in Chapter 15, the complicated concepts of Frobenius elements are introduced in chapter 16.
Z-varieties and the reciprocity law in context of Frobenius are re-explained in the penultimate chapter.
The last chapter takes off in a way to show that math is a evolving area of research.
THe book boasts excellent credentials including a preface by Barry Mazur.
March 1, 2015
Even though I probably wouldn't have understood it anyway, added with the fact that after a few chapters I stopped seriously working on any of the example problems, I felt that overall it wasn't very well explained. I sort of got the sense of what the Galois group is, but didn't really understand quadratic reciprocity. and elliptic curves. I guess it was a good jumping off point cos I found out about those things ever so dimly on my own. But it could've been better explained with more diagrams and narrative.
March 3, 2016
A rare book attempting to explain Galois theory and representation theory, to the extent of trying to illustrate how Fermat's Last Theorem and reciprocity laws are formulated. I personally find this book too difficult for laymen, though it does try to make it accessible by not skipping basic definitions and explanations.
Overall, enlightening but difficult read despite the intended reader being general audience.
Overall, enlightening but difficult read despite the intended reader being general audience.
July 17, 2009
Relatively fearless for the first 16 chapters, but by the time it gets to Frobenius and linear Galois representations to show reciprocity it becomes quite fearful. An interesting if underdeveloped incite into what mathematicians find wonderful about pure math and number theory that doesn't really achieve its goal.
October 22, 2008
Interesting stuff. The first have about p-adic numbers and all that was really interesting. The last half was way over my head though. Not a book for "lay" people, more for less-than-casual number-theory enthusiasts
November 30, 2013
Excellent introduction to Galois theory and representation theory. Not for all audiences though. Some general knowledge of maths and ideally Algebra is a must to understand the material presented in this book. Those who can approach this material will enjoy the beauty of modern mathematics.
Currently reading
December 19, 2008this is the 2nd or 3rd book I've read/am reading about this "symmetry group" stuff ... it is a particularly difficult subject for me, and I'm struggling to understand it.
October 19, 2011
Cracking book. Easy to follow and very clearly explained
September 24, 2013
Incredible! Explaining Galois representation, elliptic curves and Fermats theorem for laymen!
January 31, 2017
Try to cover very advanced topic in popular math fashion. A partial success. Part 3 of book is very very difficult to understand and is the main motivation for the book.
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